EXACT ANALYSIS OF UNSTEADY CONVECTIVE DIFFUSION FOR

Bangmod Int. J. Math. & Comp. Sci.
Vol. 1, No. 1, 2015: Pages 63 – 81
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ISSN : 2408-154X
EXACT ANALYSIS OF UNSTEADY CONVECTIVE
DIFFUSION FOR BLOOD FLOW WITH INTERPHASE
MASS TRANSFER IN MAGNETIC FIELD
Nirmala P. Ratchagar1 and R.Vijaya Kumar2,∗
1 Department of Mathematics, Annamalai University, Annamalainagar, Tamilnadu - 608 002, INDIA
E-mail : nirmalapasala@yahoo.co.in
2 Department of Mathematics, Mathematics Section, Faculty of Engineering and Technology, Annamalai
University, Annamalainagar,Tamilnadu - 608 002, INDIA
E-mail : rathirath viji@yahoo.co.in
*Corresponding author.
Abstract This paper deals with a mathematical model for exact analysis of miscible dispersion of solute
with interphase mass transfer in a blood(couple stress fluid) flow bounded by porous beds under the
influence of magnetic field. The three coefficients, namely, exchange coefficient, convection coefficient,
and dispersion coefficient are evaluated asymptotically at large-time using generalized dispersion model.
The dispersion equation is used to calculate the mean concentration distribution of a solute, bounded by
the porous layer, and is expressed as a function of dimensionless axial distance and time. It is computed
for different values of Hartmann number (M ), Couple Stress Parameter(a), reaction rate Parameter(β)
and Porous Parameter (σ).
MSC: 76S05, 76W05,76D05
Keywords: magnetic field, couple stress fluid, porous medium, generalized dispersion model.
Submission date: 11 February 2015 / Acceptance date: 12 May 2015 /Available online: 14 May 2015
c Theoretical and Computational Science and KMUTT-PRESS 2015.
Copyright 2015 1. Introduction
In many biomedical problems, the interphase mass transfer plays an important role
because many physiological situation involve interphase mass transfer. Therefore, it is
necessary to develop a technique for handling such problems, which involve interphase
mass transport. Several authors have studied various characteristics of dispersion were
mainly concerned with Taylors dispersion [23], which is valid for large time. Physiological
fluid flow problems have been mainly concerned with transient phenomena where Taylors
model is not valid. However, Sankarasubramanian and Gill[18] have developed an analytical method to analyze transient dispersion of non-uniform initial distribution called
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generalized dispersion in laminar flow in a tube with a first order chemical reaction at
the tube wall. This method can be applied to physiological problems, where a first order
chemical reaction occurs at the tube wall. One such situation is transport of oxygen and
nutrients to tissue cells and removal of metabolic waste products from tissue cells. Interphase mass transfer also takes places in pulmonary capillaries, where the carbon dioxide
is removed from the blood and oxygen is taken up by the blood.
Rudraiah et al.[17] studied the dispersion in a stokes couple-stress fluid flow by using
the generalized model of Gill and Sankarasubramanian [8]. Considering solute reaction
at the channel walls in their all time analysis of dispersion, reaction at the walls is of
practical interest and in the simplest case, a first order chemical reaction at the walls is
considered by them, in carrying out an exact analysis of unsteady convection in couple
stress fluid flows.
Gill [7] developed a local theory of Taylor diffusion in fully developed laminar tube
flow with a periodic condition at the inlet of the tube. Gill [8] obtained an exact solution
to the unsteady convective diffusion equation for miscible displacement in fully developed
laminar flow in tubes by defining dispersion coefficient to be functions of time. Shivakumar
et al. [20] have obtained a closed-form solution for unsteady convective diffusion in a fluidsaturated sparsely packed porous medium using the generalized dispersion model of [8].
In all these investigations, it is assumed that, the solute does not chemically react in the
liquid in which it is dispersed. Gupta [9], following Taylor [23], studied the effects of
homogeneous and heterogeneous reaction on the dispersion of a solute in the laminar flow
of Newtonian fluid between two parallel plates.
Shukla et al.[19], Soundalgekar [21], Meena Priya[10], Dulal Pal [3] and Dutta et al.
[4] studied dispersion in non-Newtonian fluids by considering only homogeneous firstorder chemical reaction in the bulk of the fluid. Chandra and Agarwal [2] considered
dispersion in simple microfluid flows taking only homogeneous reaction into consideration. Suvadip Paul [22] examine a the influence of angularity on the transport process
under the combined effects of reversible and irreversible wall reactions, when the flow is
driven by a pressure gradient comprising of steady and periodic components. Francesco
Gentile [5] studied the transport formulation proposed in [6] was further developed to
account for the time dependency of the problem. Prathap Kumar et al.[11] also investigated the effect of homogeneous and heterogeneous reactions on the solute dispersion
in composite porous medium. Recently,Ramana Rao[13], Ramana[14,15] and Ramana et
al [16] studied combined the effect of non-Newtonian rheology and irreversible boundary
reaction on dispersion in a Herschel-Bulkley fluid through a conduit, (pipe/channel) by
using the generalized dispersion model proposed by [8]. Pauling [12] first reported that
the erythrocytes orient with their disk plane parallel to the magnetic field.
This paper deals with the effect of couple stress and magnetic field on the unsteady
convective diffusion, with interphase mass transfer by using the generalized dispersion
model of Sankarasubramanian and Gill [18]. Convection coefficient K1 and Dispersion
coefficient K2 are influenced by the couple stress parameter arising due to suspension in
the fluid, magnetic field and porous parameter. The exchange coefficient K0 arises mainly
due to the interphase mass transfer, and it is independent of the solvent fluid velocity.
The interphase mass transfer also influence the convection and dispersion coefficients.
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2. Mathematical Formulation
We have considered a steady laminar and fully developed flow (unidirectional) in a
channel bounded by porous layers and separated by a distance 2h. A schematic diagram
of the physical configuration and the description of the initial slug input of concentration
are shown in figure 1. A uniform magnetic field Bo is applied in the y-direction to the
flow of blood. We make the following assumption for electromagnetic interactions, i) the
induce magnetic field and the electric field produced by the motion of blood are negligible
(since blood has low magnetic Reynolds number). ii) no external electric field is applied.
Flow region is divided into two sub-region such as Fluid film region and Porous tissue
region. The governing equation of the motion for flow in vector form is given by
y
B0
Region 2
y=h
Region 1
y=0
-Xs/2
x
Xs/2
Region 2
y=-h
B0
Figure 1. Physical Configuration
Region 1:Fluid Film Region
Conservation of mass for an incompressible flow
∇ · ~q = 0
(2.1)
Conservation of momentum
∂~q
ρ
+ (~q · ∇)~q = −∇p + µ∇2 ~q − λ∇4 ~q + J × B
∂t
(2.2)
Region 2:Porous Tissue Region
Conservation of mass for an incompressible flow
∇ · ~q = 0
(2.3)
Conservation of momentum
∂~q
µ
ρ
+ (~q · ∇)~q = −∇p + µ∇2 ~q − (1 + β1 )~q
∂t
k
(2.4)
Maxwell’s equations are
∇ · B = 0, ∇ × ·B = µ0 J
∇×E =−
∂B0
∂t
Ohm’s law
J = σ0 (E + ~q × B0 )
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Conservation of species
~
∂C
+ (~q · ∇)C = D∇2 C
∂t
In cartesian form, using the above equation (2.1)-(2.4) becomes
Region 1:Fluid Film Region
(2.5)
∂4u
∂p
∂2u
(2.6)
+ µ 2 − λ 4 − B02 σ0 u,
∂x
∂y
∂y
∂p
(2.7)
0=−
∂y
Region 2:Porous Tissue Region
∂p µ
0=−
− (1 + β1 )up ,
(2.8)
∂x k
∂p
0=−
(2.9)
∂y
where, ~q = ~iu + ~jv, u is the x components of velocity, p is the pressure, ρ is the density
of the fluid, µ is the viscosity of the fluid, λ is the couple stress parameter, k is the
permeability of the porous medium, E the electric field,σ0 is the electrical conductivity,
J the current density and up is the darcy velocity. It may be noted that, (2.8) is the
modified darcy equation. where, β1 is the couple stress parameter.
We consider the dispersion of reactive solute in the fully developed flow through a
parallel plate channel bounded by porous beds. Introduced a slug of concentration C =
C0 ψ1 (x)Y1 (y). The mass balance equation (2.5) concerning the solute concentration C
undergoes heterogeneous chemical reaction such as
2
∂C
∂C
∂ C
∂2C
+u
=D
+
(2.10)
∂t
∂x
∂x2
∂y 2
0=−
where, D is the molecular diffusivity.
The boundary conditions on velocity are
−α
∂u
= √ (u − up ) at y = h,
(2.11)
∂y
k
α
∂u
= √ (u − up ) at y = −h,
(2.12)
∂y
k
∂2u
= 0 at y = ±h.
(2.13)
∂y 2
where, α is the slip parameter. Eqs. (2.11) and (2.12) is Beavers and Joseph [1] slip
condition at the lower and upper permeable surfaces and equation (2.13) specifies the
vanishing of the couple stress.
Initial and Boundary conditions on concentration
The initial distribution assumed to be in a variable separable form is given by
C(0, x, y) = C0 ψ1 (x)Y1 (y),
(2.14)
The heterogeneous reaction conditions are:
−D ∂C
∂y
D ∂C
∂y
=
=
Ks C
Ks C
at y = h and
at y = −h
)
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where, Ks is the reaction rate constant catalysed by the walls and C0 is a reference
concentration.
As the amount of solute in the system is finite,
C(t, ∞, y)
=
∂C
(t, ∞, y) = 0
∂x
(2.16)
where, C0 is reference concentration.
Now, we introduce the non-dimensional quantities,
U=
y
C
Dx
Dxs
Dt
h¯
u
u
; η= ; θ=
; X = 2 ; Xs = 2 ; τ = 2 ; P e =
;
u
¯
h
C0
h u
¯
h u
¯
h
D
Equation (2.6) and (2.10) in non-dimensional form as
∂4U
∂2U
− a2 2 + a2 M 2 U = P a2
4
∂η
∂η
(2.17)
and
∂θ
∂θ
∂2θ
1 ∂2θ
+ U∗
+
=
2
∂τ
∂X1
P e2 ∂X1
∂η 2
q
2 ∂p
λ
h
2
where, P = −h
,
l
=
µ ∂x
µ and a = l is the couple stress parameter, M =
∗
u
¯h
D
(2.18)
B02 σ0 h2
µ
is the
¯
U −U
¯
U
is the peclet number, U =
is normalized
square of the Hartmann number, P e =
axial component of velocity as x1 = x − u
¯t which is dimensionless form is X1 = X − τ
where X1 = xh12D
a
The initial and boundary conditions of (2.11) to (2.16) in dimensionless form
∂U
= −ασ(U − Up ) at η = 1
∂η
(2.19)
∂U
= ασ(U − Up ) at η = −1
∂η
(2.20)
∂2U
= 0 at y = ±1
∂η 2
(2.21)
where, σ =
√h
k
is the porous parameter.
θ(0, X, η)
∂θ
∂η
∂θ
∂η
=
=
= ψ(X)Y (η),
−βθ
βθ
θ(τ, ∞, η)
at η = 1,
at η = −1
=
(2.22)
)
∂θ
(τ, ∞, η) = 0
∂X
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3. Method of solution
Velocity distribution
The solution of Equation (2.17) can be written as:
P
(3.1)
M2
where, C1 ,C2 ,C3 and C4 are constants. Applying the boundary conditions (2.19)-(2.21)
in (3.1), we obtain the the velocity of blood as
P
U (η) = 2C1 Coshm1 η + 2C3 Coshm3 η + 2
(3.2)
M
The normalized axial components of velocity obtained from(3.1) is
¯
U −U
2
C1 Sinhm1
C3 Sinhm3
=
+
C
Coshm
η
+
C
Coshm
η
−
U∗ =
1
1
3
3
¯
A1
m1
m3
U
(3.3)
U (η) = C1 em1 η + C2 e−m1 η + C3 em3 η + C4 e−m3 η +
where,
¯=1
U
2
Z1
U (η)dη =
−1
2C1 Sinhm1
2C3 Sinhm3
P
+
+ 2
m1
m3
M
(3.4)
Generalized Dispersion model
The solution of (2.18) subject to the conditions (2.22)-(2.24), following Gill and Sankarasubramanian [18] is
θ(τ, X, η) =
∞
X
fk (τ, η)
k=0
∂ k θm
,
∂X k
(3.5)
where, θm is the dimensionless cross sectional average concentration and is given by
θm
1
=
2
Z1
θ(τ, X, η)dη
(3.6)
−1
Equation (2.18) is multiplied throughout by
the limits -1 to 1 and using (3.6) we get,
1
2
and integrated with respect to y between
1
Z1
∂θm
1 ∂ 2 θm
1 ∂θ
1 ∂
= 2
+
−
U ∗ θdη
∂τ
Pe ∂X 2
2 ∂η −1 2 ∂X
(3.7)
−1
Using Eq. (3.5) in (3.7), we get the dispersion model for θm as
1
∂θm
1 ∂ 2 θm
1 ∂
∂θm
=
+
(f0 (τ, η)θm (τ, X) + f1 (τ, η)
(τ, X) + . . .)
∂τ
Pe2 ∂X 2
2 ∂η
∂X
−1
1 ∂
−
2 ∂X
Z1
U ∗ (f0 (τ, η)θm (τ, X)
−1
+f1 (τ, η)
∂ 2 θm
∂θm
(τ, X) + f2 (τ, η)
(τ, X) + . . .)dη
∂X
∂X 2
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The generalized dispersion model of [7] is defined as
∞
X
∂θm
∂ i θm
=
Ki (τ )
∂τ
∂X i
i=0
(3.9)
substituting (3.9) in (3.8) we get,
∂θm
∂ 2 θm
K0 θm + K1
+ K2
∂X
∂X 2
=
1
1 ∂ 2 θm
1 ∂
∂θm
+
(f0 θm + f1
+ . . .)
Pe2 ∂X 2
2 ∂η
∂X
−1
1
Z
∂θm
1 ∂
U ∗ f0 (τ, ξ)θm (τ, X) + f1 (τ, ξ)
(τ, X)
−
2 ∂X
∂X
−1
∂ 2 θm
+f2 (τ, ξ)
(τ, X) + . . . dη
∂X 2
2
m ∂ θm
Equating like coefficient of θm , ∂θ
∂X , ∂X 2 . . . we get Ki ’s are,
δi2
1 ∂fi
1
Ki (τ ) = 2 +
(τ, 1) −
Pe
2 ∂η
2
Z1
fi−1 (τ, η)U ∗ (τ, η)dη
(3.10)
−1
(i = 1, 2, 3, . . .)
where, f−1 = 0 and δi2 is the Kroneckar delta defined by
δi2 =
1,
0,
i=j
i 6= j
The exchange coefficient K0 (τ ) accounts for the non-zero solute flux at the channel
wall, and negative sign indicates the depletion of solute in the system with time caused
by the irreversible reaction, which occurs at the channel wall. The presence of non-zero
solute flux at the walls of the channel, also affects the higher order Ki due to the explicit
i
appearance of ∂f
∂η (τ, 1) in equation (3.10). Equation (3.9) can be truncated after the term
involving K2 without causing serious error, because K3 , K4 , etc. become negligibly small
compared to K2 . The resulting model for the mean concentration is
∂θm
∂θm
∂ 2 θm
= K0 (τ )θm + K1 (τ )
+ K2 (τ )
∂τ
∂X
∂X 2
(3.11)
To solve the equation (3.11), we need the coefficients Ki (τ ) in addition to the appropriate initial and boundary conditions. For this, the corresponding function fk must be
determined. So, substituting (3.5) into (2.18), the following set of differential equations
for fk are generated.
k
X
∂fk
∂ 2 fk
1
∗
=
−
U
f
+
f
−
Ki fk−i ,
k−1
k−2
∂τ
∂η 2
Pe2
i=0
(k = 0, 1, 2, . . .)
(3.12)
where, f−1 = f−2 = 0.
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To evaluate Ki0 s, we need to know the fk ’s which are obtained by solving (3.12) for
fk ’s subject to the boundary conditions,
fk (τ, 0) =
∂fk
(τ, 1) =
∂η
∂fk
(τ, 0) =
∂η
1
2
f inite,
(3.13)
−βfk (τ, 1),
(3.14)
0,
(3.15)
δk0 , (k = 0, 1, 2)
(3.16)
Z1
fk (τ, η)dη
=
−1
The function f0 and the exchange coefficient K0 are independent of the velocity and
can be solved easily. It should be pointed out here, that, a simultaneous solution has to
be obtained from these two quantities since K0 , which can be obtained from (3.10) as
1
1 ∂f0
(3.17)
K0 (τ ) =
2 ∂η −1
Substituting k = 0 in equation (3.12) we get the differential equation for f0 as
∂f0
∂ 2 f0
=
− f0 K 0
(3.18)
∂τ
∂η 2
We derive an initial condition for f0 using (3.6) by taking τ = 0 in that equation to get
1
θm (0, X) =
2
Z1
θ(0, X, η)dη
(3.19)
−1
Substituting τ = 0 in (3.5) and setting fk (η) = 0(k = 1, 2, 3) gives us the initial condition
for f0 as
θ(0, X, η)
f0 (0, η) =
(3.20)
θm (0, X)
We note that the left hand side of (3.20) is a function of η only and the right hand side is
a function of both X and η. Thus, clearly the initial concentration distribution must be
a separable function of X and η. Substituting equation (2.14) and (3.19) into equation
(3.20), we get
f0 (0, η) =
1
2
ψ(η)
R1
ψ(η)dη
(3.21)
−1
The solution of the reaction diffusion equation (3.18) with these conditions may be formulated as
 τ

Z
f0 (τ, η) = g0 (τ, η) exp − K0 (η)dη 
(3.22)
0
from which it follows that g0 (τ, η) has to satisfy
∂g0
∂ 2 g0
=
∂τ
∂η 2
(3.23)
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with conditions
f0 (0, η)
= g0 (0, η) =
1
2
ψ(η)
,
R1
ψ(η)dη
(3.24)
−1
g0 (τ, 0) = f inite,
∂g0
(τ, 1) = −βg0 (τ, 1).
∂η
(3.25)
(3.26)
The solution of (3.23) subject to conditions (3.24)-(3.26) is
g0 (τ, η) =
∞
X
An Cos(µn η)exp[−µ2n τ ]
(3.27)
n=0
where, µn ’s are the roots of
µn tanµn = β,
n = 0, 1, 2, . . .
(3.28)
and An ’s are given by
2
An = R1
ψ(η)Cosµn ηdη
−1
1+
Sin2µn
2µn
R1
(3.29)
ψ(η)dη
−1
from (3.22), it follows that
9
P
f0 (τ, η) =
2g0 (τ, η)
R1
g0 (τ, η)dη
=
n=0
9
P
An exp[−µ2n τ ]Cosµn η
n=0
−1
(3.30)
An
2
µn exp[−µn τ ]Sinµn
The first ten roots of the transcendental equation (3.28) are obtained using MATHEMATICA 8.0 and are given in Table 1. We find that these ten roots ensured convergence
of the series seen in the expansions for f0 and K0 . Having obtained f0 , we get K0 from
(3.17) in the form
9
P
K0 (τ ) =
An µn exp[−µ2n τ ]Sinµn
− n=09
P
n=0
(3.31)
An
2
µn exp[−µn τ ]Sinµn
K0 (τ ) is independent of velocity distribution.
As τ → ∞, we get the asymptotic solution for K0 from (3.31) as
K0 (∞) = −µ20
(3.32)
where, µ0 is the first root of the equation (3.28). Physically, this represents first order
chemical reaction coefficient having obtained K0 (∞). We can now get K1 (∞), from (3.10)
(with i = 1) knowing f0 (∞, η) and f1 (∞, η). Likewise, K2 (∞), K3 (∞), . . . require the
knowledge of K0 , K1 , f0 , f1 and f2 . Equation (3.30) in the limit τ → ∞ reduces to
µ0
f0 (∞, η) =
Cos(µ0 η)
(3.33)
Sinµ0
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Then we find f1 , K1 , f2 , and K2 . For asymptotically long times, i.e., τ → ∞, equation
(3.10) and (3.12) give us Ki ’s and fk ’s as
Z1
δi2
− βfi (∞, 1) − U ∗ fi−1 (∞, η)dη, (i = 1, 2, 3)
Ki (∞) =
Pe2
−1
2
d fk
1
2
∗
+
µ
f
=
(U
+
K
)f
−
−
K
1 k−1
2 fk−2 , (k = 1, 2)
0 k
dη 2
Pe2
(3.34)
(3.35)
The fk ’s must satisfy the conditions (3.6) and this permits the eigen function expansion
in the form of
9
X
Bj,k Cos(µj η), k = 1, 2, 3, . . .
(3.36)
fk (∞, η) =
j=0
Substituting (3.36) in (3.35) and multiplying the resulting equation by Cos(µj η) and
integrating with respect
" to η from -1 to 1,
∞
P
1
1
Bj,k Cosµj η = µ2 −µ
Bj,k−2 Cosµj η
2
2
Pe
0
j
j=0
#
∞
∞
P
P
−U ∗
Bj,k−1 Cosµj η −
Ki Bj,k−i Cosµj η
j=0
j=0
multiplying by Cosµl η and integrating with respect to η, we get


−1 X
k
9
X
1
1
Sin2µ
j
 Bj,k−2 −
Bj,k =
Ki Bj,k−i − 1 +
Bj,k−1 I(j, l)
(µ2j − µ20 ) Pe2
2µj
i=1
j=0
k = (1, 2)
(3.37)
where,
Z1
I(j, l)
=
U ∗ Cosµj ηCosµl ηdη = I(l, j)
(3.38)
−1
Bj,−1
=
0, Bj,0 = 0 f or j = 1 to 9
(3.39)
The first expansion coefficient B0,k in equation (3.36) using conditions (3.13)-(3.16) can
be expressed in terms of Bj,k (j = 1 to 9) as,
fk
= B0,k Cosµ0 η +
∞
X
Bj,k Cosµj η (from equation 3.36)
j=1
By integrating this equation, we get
∞
Sinµ0 η X
Sinµj η
0 = B0,k
+
Bj,k
µ0
µj
j=1
(Using the boundary condition
R1
fk (τ, η)dη = δk0 = 0)
−1
B0,k
= −
µ0
Sinµ0
X
9
j=1
Bj,k
Sinµj
, (k = 1, 2, 3, . . .)
µj
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Further, from (3.36) and (3.33) we find that
µ0
(3.41)
Sinµ0
Substituting i = 1 in (3.34) and using (3.38), (3.39) and (3.41) in the resulting equation,
we get
I(0, 0)
i
(3.42)
K1 (∞) = − h
0
1 + Sin2µ
2µ0
B0,0
=
Substituting i = 2 in (3.34) and using (3.36), (3.38) and (3.41) in the resulting equation,
we get
9
X
1
Sinµ0
Bj,1 Ij,0
−
2
Pe
0
µ0 1 + Sin2µ
j=0
2µ0
−1
µ0
0
= −(µ2j − µ20 )−1 1 + Sin2µ
2µ0
Sinµ0 I(j, 0)
K2 =
where, Bj,1
(3.43)
Using the asymptotic coefficients K0 (∞), K1 (∞), and K2 (∞), in (3.9), one can determine the mean concentration distribution as a function of X, τ and the parameters a
and β.
The initial condition for solving (3.9) can be obtained from (2.22) by taking the crosssectional average. Since we are making long time evaluations of the coefficients, we note
that the side effect is independent of θm on the initial concentration distribution. The
solution of (3.9) with asymptotic coefficients can be written as:
"
#
2
1
[X + K1 (∞)τ ]
p
θm (τ, X) =
exp K0 (∞)τ −
(3.44)
4K2 (∞)τ
2Pe πK2 (∞)τ
where,
θm (τ, ∞) = 0,
∂θm
(τ, ∞) = 0
∂X
4. Results and Discussions
We have modeled the solvent as a couple stress fluid(blood) and studied dispersion of
solute in a blood flow bounded by porous beds in the presence of magnetic field considering
heterogeneous chemical reaction, on the interphase. The walls of the channel act as
catalysts to the reaction. The problem brings into focus three important dispersion
coefficients namely, the exchange coefficient −K0 which arises essentially due to the wall
reaction, the convective coefficient −K1 and diffusive coefficient K2 . The asymptotic
values of these three coefficient are plotted in figures 2 to 16 for different value of Hartmann
number (M < 1, M = 1, M > 1), Couple Stress Parameter(a = 5, 10, 20), reaction rate
Parameter(β = 10−2 , 1, 102 ) and Porous Parameter (σ = 100, 200, 500). This paper gives
the solutions in MATHEMATICA 8.0.
From figure 2, it is cleat that −K0 increases with an increase in the wall reaction
parameter(β) but it is unaffected by the couple stress parameter(a), Hartmann number
(M ) and porous parameter (σ). The classical convective coefficient(−K1 ) is plotted in
figures 3 to 5 versus wall reaction parameter (β) for different values of couple stress
parameter(a), Hartmann number (M ) and porous parameter (σ) respectively for a fixed
value of slip parameter α = 0.1, β1 = 0.1, P e = 100 and h = 2. From figure 3 and 4,
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we conclude that −K1 decrease, with increasing values of couple stress parameter and
Hartmann number. Figure 5 shows that increase in (−K1 ), increase the value of (σ). This
is advantageous is maintaining the laminar flow.
The scaled dispersion coefficient K2 − Pe−2 is plotted versus β in figures 6 to 8 for
different values of a, M and σ. Figure 6 and 7, show that the increase in a and M, the
effective dispersion coefficient decreases. From figure 8, we conclude that, increase in σ
is to increase the effective dispersion coefficient K2 . These are useful in the control of
dispersion of a solute.
The cross sectional average concentration θm is plotted versus X in figures 9 to 12
respectively for different values of a, M, σ and β and for fixed values of other parameters
given in these figures. It is shows that increase in β and σ decreases θm , while an increase
in (a) and (M ) increases θm as expected on the physical ground. This may be attributed
to increase in σ and β is to reduce the velocity and hence to decrease θm .
The cross sectional average concentration θm is also plotted against the time τ in
Figures 13 to 16 respectively for different values of a, M, σ and β and for fixed values of
other parameters given in these figures. We conclude that the peak of θm decreases with
an increase in β occurring at the lower interval of time τ . We also note that, although
the peak decreases with an increase in σ and increases with an increase in a and M, but
occurs at almost at the same interval of time τ.These results are useful to understand the
transport of solute at different times.
5. Conclusion
The present investigation brings out some interesting results on the dispersion process
in flows of blood modeled as couple stress fluid in the presence of magnetic field. The
convective dispersion process is analyzed employing the dispersion model of Gill and
Sankarasubramanian [8]. It is observed that the effect of magnetic field on dispersion
coefficient is found to decrease with increase in β and cross sectional average concentration
is to increase the time to reach its peak value.
References
[1] G.S. Beavers and D. D. Joseph,Boundary conditions at a naturally permeable wall,
J. Fluid Mech. 30(1967),197–207.
[2] P. Chandra and R. P. Agarwal,Dispersion in simple micro fluid flows, International
Journal of Engineering Science 21(1983), 431–442.
[3] Dulal Pal, Effect of chemical reaction on the dispersion of a solute in a porous
medium, Applied Mathematical Modeling 23,(1999), 557–566.
[4] B. K. N. Dutta , C. Roy and A. S. Gupta, Dispersion of a solute in a nonNewtonian fluid with simultaneous chemical reaction, Mathematica-Mechanical fasc.,
2(1974),78–82.
[5] Francesco Gentile and Paolo Decuzzi, Time dependent dispersion of nanoparticles in
blood vessels, J. Biomedical Science and Engineering 3(2010),517–524.
[6] F. Gentile, M. Ferrari and P. Decuzzi,The transport of nanoparticles in blood vessels:
The effect of vessel permeability and blood rheology, Annals of Biomedical Engineering2(36),(2008) , 254–261.
[7] W. N. Gill,A note on the solution of transient dispersion problems, Proc. Roy. Soc.
Lond. A 298(1967) ,335–339.
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[8] W. N. Gill and R. Sankarasubramanian, Exact analysis of unsteady convective diffusion, Proc. Roy. Soc. Lond. A 316(1970) , 341–350.
[9] P. S. Gupta and A. S. Gupta,Effect of homogeneous and heterogeneous reactions on
the dispersion of a solute in the laminar flow between two plates, Proc. Roy. Soc.
Lond. A 330(1972),59–63.
[10] P. Meena Priya and Nirmala P. Ratchagar,Generalized dispersion of atmospheric
aerosols on unsteady convective diffusion in couple stress fluid bounded by electrodes,
Int. Journal of Applied Mathematics and Engineering Sciences 5(1)(2011),59–72.
[11] J. Prathap Kumar, J. C. Umavathi and Shivakumar Madhavarao, Effect of homogeneous and heterogeneous reactions on the solute dispersion in composite porous
medium, International Journal of Engineering, Science and Technology, Vol. 4, No.
2(2012), 58–76.
[12] L. Pauling, C D. Coryell, The magnetic Properties and Structure of Hemoglobin, Oxyhemoglobin and Carbonmonoxy Hemoglobin, In: Proceedings of the National Academy of Science, USA 22 (1936), 210-216.
[13] V. V. Ramana Rao, and D. Padma, Homogeneous and heterogeneous reaction on the
dispersion of a solute in MHD Couette flow II, Curr. Sci., 46(1977), 42–43.
[14] B. Ramana and G. Sarojamma, Unsteady Convective Diffusion in a Herschel-Bulkley
Fluid in a Conduit with Interphase Mass Transfer, International Journal of Mathematical Modelling and Computations, Vol. 02, No. 03 (2012), 159 –179.
[15] B. Ramana and G. Sarojamma, Effect of Wall Absorption on dispersion of a solute
in a Herschel-Bulkley Fluid through an annulus, Pelagia Research Library ,Advances
in Applied Science Research, 3 (6) (2012), 3878–3889.
[16] B. Ramana, G. Sarojamma, B. Vishali and P. Nagarani,Dispersion of a solute in
a Herschel-Bulkley fluid flowing in a conduit, Journal of Experimental Sciences ,
3(2),(2012), 14–23.
[17] N. Rudraiah, Dulal Pal and P. G. Siddheswar, Effect of couple stress on the unsteady
convective diffusion in fluid flow through a channel, BioRheology, Vol.23(1986), 349–
358.
[18] R. Sankarasubramanian and W. N. Gill,Unsteady convective diffusion with interphase
mass transfer, Proc. Roy. Soc. London, A 333 (1973), 115–132.
[19] J. B. Shukla, R. S. Parihar and B. R. P. Rao, Dispersion in non-Newtonian fluids:
Effects of chemical reaction, Rheologica Acta,18(1979),740–748.
[20] P. N. Shivakumar, N. Rudraiah, D. Pal and P. G. Siddheshwar,Closed form solution
for unsteady diffusion in a fluid saturated sparsely packed porous medium, Int. Comm.
Heat Mass Transfer, 14 (1987), 137–145.
[21] V. M. Soundalgekar and P. Chaturani, Effects of couple-stresses on the dispersion of
a soluble matter in a pipe flow of blood, Rheologica Acta, 19(1980),710–715.
[22] Suvadip Paul and B. S. Mazumder, Transport of reactive solutes in unsteady annular
flow subject to wall reactions, European Journal of Mechanics B/Fluids, 28(2009),
411–419.
[23] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube,
Proc. Roy. Soc. Lond. A., 219 (1953),186–203.
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Table 1. Roots of the equation µn tanµn = β
β
10−2
0.05
10−1
0.5
1.0
5.0
10.0
100.0
µ0
0.099834
0.22176
0.311053
0.653271
0.860334
1.31384
1.42887
1.55525
µ1
3.14477
3.15743
3.1731
3.29231
3.42562
4.03357
4.3058
4.66577
µ2
6.28478
6.29113
6.29906
6.36162
6.4373
6.9096
7.22811
7.77637
µ3
9.42584
9.43008
9.43538
9.47749
9.52933
9.89275
10.2003
10.8871
µ4
12.5672
12.5703
12.5743
12.606
12.6453
12.9352
13.2142
13.9981
µ5
15.7086
15.7111
15.7143
15.7397
15.7713
16.0107
16.2594
17.1093
µ6
18.8501
18.8522
18.8549
18.876
18.9024
19.1055
19.327
20.2208
µ7
21.9916
21.9934
21.9957
22.0139
22.2126
22.2126
22.4108
23.3327
µ8
25.1331
25.1347
25.1367
25.1526
25.1724
25.3276
25.5064
26.445
µ9
28.2747
28.2761
28.2779
28.292
28.3096
28.4483
28.6106
29.5577
-K0
2.0
1.5
1.0
0.5
Β
2
4
6
8
10
Figure 2. Plots of exchange coefficient versus reaction rate parameter β
-K1
1.10
1.08
a=5
1.06
1.04
a=20
1.02
Β
20
40
60
80
100
Figure 3. Plots of convective coefficient −K1 versus β for different values of a when M = 1
and σ = 100
-K1
1.10
M=1
1.08
M=1.5
1.06
1.04
M=2
1.02
Β
20
40
60
80
100
Figure 4. Plots of convective coefficient −K1 versus β for different values of M when σ = 100
and a = 5
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-K1
1.12
Σ=100
1.10
1.08
Σ=200
1.06
1.04
Σ=500
1.02
Β
20
40
60
80
100
Figure 5. Plots of convective coefficient −K1 versus β for different values of σ when M = 1
and a = 5
1
K2 -
Pe2
a=5
0.00135
a=10
0.00130
a=20
0.00125
Β
0.2
0.4
0.6
0.8
1.0
Figure 6. Plots of scale dispersion coefficient K2 − P e−2 versus β for different values of a
when M = 1 and σ = 100
1
K2 -
Pe2
0.0020
M=0.5
0.0015
M=1.0
0.0010
M=1.5
Β
0.2
0.4
0.6
0.8
1.0
Figure 7. Plots of scale dispersion coefficient K2 − P e−2 versus β for different values of M
when σ = 100 and a = 5
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1
K2 -
Pe2
0.0017
Σ=100
0.0016
Σ=200
0.0015
Σ=500
0.0014
0.0013
Β
0.2
0.4
0.6
0.8
1.0
Figure 8. Plots of scale dispersion coefficient K2 − P e−2 versus β for different values of σ
when M = 1 and a = 5
HPeLΘm
8
a=5
6
a=10
a=20
4
2
X
0.2
0.4
0.6
0.8
1.0
Figure 9. Plots of mean concentration θm versus X for different values of a when M = 1, σ =
100, β = 10−2 and τ = 0.6
HPeLΘm
14
12
10
M=0.5
8
M=1.0
6
M=1.5
4
2
X
0.2
0.4
0.6
0.8
Figure 10.Plots of mean concentration θm versus X for different values of M when a = 5, σ =
100, β = 10−2 and τ = 0.6
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HPeLΘm
8
Σ=100
6
Σ=200
4
Σ=500
2
X
0.2
0.4
0.6
0.8
1.0
Figure 11. Plots of mean concentration θm versus X for different values of σ when M =
1, a = 5, β = 10−2 and τ = 0.6
HPeLΘm
8
Β=10-2
6
Β=1
4
Β=102
2
X
0.2
0.4
0.6
0.8
Figure 12. Plots of mean concentration θm versus X for different values of β when M =
1, a = 5, σ = 100 and τ = 0.6
HPeLΘm
8
6
a=5
a=10
4
a=20
2
Τ
0.2
0.4
0.6
0.8
1.0
Figure 13.Plots of mean concentration θm versus τ for different values of a when M = 1, σ =
100, β = 10−2 and X = 0.8
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HPeLΘm
12
10
M=0.5
8
M=1.0
6
M=1.5
4
2
Τ
0.2
0.4
0.6
0.8
1.0
Figure 14.Plots of mean concentration θm versus τ for different values of M when a = 5, σ =
100, β = 10−2 and X = 0.8
HPeLΘm
8
6
Σ=100
Σ=200
4
Σ=500
2
Τ
0.2
0.4
0.6
0.8
1.0
Figure 15.Plots of mean concentration θm versus τ for different values of σ when M = 1, a =
5, β = 10−2 and X = 0.8
HPeLΘm
8
6
Β=10-2
Β=1
4
Β=102
2
Τ
0.2
0.4
0.6
0.8
1.0
Figure 16. Plots of mean concentration θm versus τ for different values of β when M = 1, a =
5, σ = 100 and X = 0.8
APPENDIX A
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p
√
a2 − a4 − 4a2 M 2
√
,
2
p
√
a2 + a4 − 4a2 M 2
√
m3 =
,
2
m1
a3 = (m1 +ασ)e ,
a4 = (m1 −ασ)e−m1 ,
a5 = (m3 +ασ)em3 ,
a6 = (m3 −ασ)e−m3 ,
P
k
∂p
a7 =
−
ασ,
2
M
µ (1 + β1 ) ∂x
2 m1
2 −m1
a8 = m1 e , a9 = m1 e
a10 = m23 em3 , a11 = m23 e−m3 ,
Eq.(A.1)
m1 =
Eq.(A.2)
Eq.(A.3)
Eq.(A.4)
Eq.(A.5)
Eq.(A.6)
Eq.(A.7)
Eq.(A.8)
Eq.(A.9)
−a7 a10 − a7 a11
,
a5 a8 − a6 a8 + a5 a9 − a6 a9 − a3 a10 + a4 a10 − a3 a11 + a4 a11
a7 a8 + a7 a9
= C4 =
,
a5 a8 − a6 a8 + a5 a9 − a6 a9 − a3 a10 + a4 a10 − a3 a11 + a4 a11
P
2C1 sinh m1 2C3 sinh m3
=
+
+ 2,
m1
m3
M
C1 sinh m1 C3 sinh m3
=
+
,
m1
m3
C1 sinh m1 C3 sinh m3
=
+
.
m31
m33
C1 = C2 =
Eq.(A.10)
C3
Eq.(A.11)
A1
A2
A3
Eq.(A.12)
Eq.(A.13)
Eq.(A.14)
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